5.3 Hilbert's Axioms
Modified for
PlaneElliptic GeometryI have tried to avoid long
numerical computations, thereby following Riemann's postulate that proofs should
be given through ideas and not voluminous computations.
—David Hilbert (1862–1943)

Introductory Notes.Consider
three distinct collinear points A, B, and C in the Riemann
Sphere model. Depending on where we start, we could place the points in any of
the following orders: A-B-C, A-C-B, B-A-C, B-C-A, C-A-B, and C-B-A.
Hence, we cannot say one of the points is between the other two points.
However, for four distinct collinear points A, B, C, and D, we
could say that two of them separate the other two. To cover this
problem, we replace the Axioms of Order with the Axioms of Separation. For
example, in the diagram on the right, point A and B separate
points C and D.
In order to use the Riemann Sphere model with the
following axiom set, we identify each pair of antipodal points as a single point
since two distinct lines are incident with a unique point, i.e., the Modified
Riemann Sphere model.

Undefined Termspoint,
line, lie,
separate, and
congruence.

Group I. Axioms of Incidence

I.1. For every two distinct points A
and B, there exists a line m
that contains each of the points A and B.
I.2. For every two distinct points A
and B, there is no more
than one line m that contain each of the points Aand B.
I.3. There exist at least two distinct points on a line. There
exist at least three points that do not lie on a line.
I.4. (Elliptic Parallel Postulate) For any
two distinct lines m and n, there exists a unique point A
contained by lines m and n.

Group II. Axioms of Separation (or Axioms of Order)

II.1. For any three distinct points A, B
and C on a line m, there is a point D on m
such that A and B separates C and D
(denoted (A, B | C, D). II.2. If (A, B | C, D), then the points A,
B, C, and D are distinct and collinear.II.3. If (A, B
| C, D), then (B, A | C, D) and (C, D |
A, B).
II.4. If (A, B | C, D), then A and C
does not separate B and D.
II.5. For any four distinct
collinear points A, B, C, and D, (A, B | C, D)
or (A, C | B, D) or (A, D | B, C).II.6. For any five distinct collinear points A,
B, C, D, and E, if (A, B | D, E), then
either (A, B | C, D) or (A, B | C, E).
II.7. Separation of
points is invariant under a
perspectivity, i.e., if (A, B | C, D)andthere is a perspectivity mapping A, B, C, and D on
line p tothe corresponding points A', B', C',
and D' on line p', then (A', B' | C', D').

Since in neutral geometry the definitions of segments,
rays, angles, and triangles all depended on betweenness of points, the
definitions all need to be revised based on the separation of points.

Exercise 5.1.1. Use the concepts from the separation axioms to
write definitions for each of the following.

(a) segment
(b) ray (Note that in the spherical model a ray is a line.)
(c) angle (Note that at the vertex of an angle there are four
angles. Consider how to distinguish which is the particular angle desired.)
(d) triangle (Note that three noncollinear points determine more
than one triangle.)

Group III. Axioms of Congruence

III.1. If A, B are two points on a line m, and A'
is a point on the same or on another line m' then it is always possible
to find a point B' on a given side of the line m' through A'
such that the segment AB is congruent to the segment A'B'(denotedAB
≅ A'B').
III.2. Congruence of segments is an
equivalence relation.
III.3. If two segments are congruent, then their
mutually complementary segments are congruent.
III.4. On the line m let AB and BC be two segments
which except for B have no point in common. Furthermore, on the same or
on another line m' let A'B' and B'C' be two segments
which except for B' also have no point in common. In that case, if
AB≅
A'B' and BC≅ B'C', then AC≅
A'C'.III.5. Let ∠(h,k) be an angle and m' a line andlet a definite side of m'. Let h' be a ray on the line m' that emanates from the
point O'. Then there exists
one and only one ray k' such that the angle ∠(h,k)
is congruent to the angle ∠(h',k') and at
the same time all interior points of the angle ∠(h',k')
lie on the given side of m'.Symbolically ∠(h,
k) ≅∠(h',k').Every angle is congruent to itself.
III.6. If for two triangles ABC and A'B'C' the congruences
AB≅A'B', AC≅A'C',∠BAC≅∠B'A'C' hold, then the congruence∠ABC≅∠A'B'C'
is also satisfied.

Group IV. Axiom of Parallels - Elliptic
geometry has no parallel lines and hence the elliptic parallel postulate
should probably be called the elliptic no
parallels postulate.

See Group I. Axioms of Incidence - I.4. (Elliptic Parallel Postulate) For any
two distinct lines m and n, there exists a unique point A
contained by lines m and n.

Group V. Axiom of Continuity

V. (Dedekind's Axiom)
Assume the set of all points on a segment AB is
the union, S1US2, of two nonempty subsets of segment AB such
that no point of S1 separates two points of S2.
Then there is a unique point C in the interior of segment AB
such that (A, C | P1, P2)
if and only if P1 is in S1 and P2
is in S2 and C is neither P1
nor P2.

Defined Terms

Consider two distinct points A
and B on a line m, then there are points C and
D on line mwhere (A, B | C, D).
The union of the set of
the two points A and B
and the set of all points X where (A, B | C, X)
is called
segmentAB/C.
The points A and B are called the
endpoints of the segment AB/C.
The set of all points X where (A, B | C, X) is
called
interior of segment AB/C. The set of all points Y
where (A, B | Y, D) is called
exterior
of segment AB/C. The segment AB/C and segment
AB/D are calledmutually complementary
segments. Note that three points are often needed to name
a segment since points A and B define two mutually
complementary segments, segment AB/C and segment AB/D.
Further note that the interior of segment AB/C is the exterior of
segment AB/D.

Let A, A', O, B be four points of a line m such that Oand B do not separate A and
A'.
The points A, A' are then said to lie on the line mon one
and the same side of the point O and the points A, B are said to
lie on the line m on different sides of the point O. The totality of
the points of the line m that lie on the same side of O is
called a ray emanating from O.The point O is
the vertex of
the ray emanating from O and A
is arelative pointof the ray
emanating from O.

Let hand kbe any two distinct rays emanating from Oand lying on distinct lines.
Further, let AB be a segment with A
and B relative points of rays h and k,
respectively, distinct from O. The pair of rays h, k is called an
angle and is denoted by ∠(h,k)
or ∠(k,h)or ∠AOB.
Let C be an exterior point of segment AB. All points that
lie in the interior of segments XY/Z where X and Y
are relative points of h and k, respectively,
and Z is a point on line OC are said to lie in the
interior of the angle ∠(h,k)
or∠AOB.

Let AB, BC, and CA be three segments
whereA, B,and C are three points which do not lie on the same line.The union of
the three segments AB, BC,and CAis
a trianglecalled
triangle ABC. (Note that
the noncollinear points A, B, and C define more than one triangle.)

Exercise 5.1.2.(a) Prove a
segment has at least three points.
(b) Prove a segment has infinitely many points.

Exercise 5.1.3.
Show that the Riemann Sphere where each pair of antipodal points is a point
satisfies each of the axioms.